A partition P1 is called a refinement of the partition P2 if every set in P1 is a subset of one of the sets in P2.

S1: The partition of the set of bit strings of length 16 formed by equivalence classes of bit strings that agree on the last eight bits is a refinement of the partition formed from the equivalence classes of bit strings that agree on the last four bits

S2: The partition formed from congruence classes modulo 6 is a refinement of the partition formed from congruence classes modulo 3

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First, try to understand what is refinement.

A partition P1 is called a refinement of the partition P2 if every set in P1 is a subset of one of the sets in P2.

S1: The partition of the set of bit strings of length 16 formed by equivalence classes of bit strings that agree on the last eight bits is a refinement of the partition formed from the equivalence classes of bit strings that agree on the last four bits

Here in P1 for each distinct pattern of last 8 bits one partition is created. In P2 for each distinct pattern of last 4 bits one partition is created.
Every partition of P1 is a subset of some partition of P2, so P1 is refinement of P2.
In same way in S2 too P1 is refinement of P2.